Graph Theory is a mathematical theory that deals with the study of graphs which are mathematical structures that are used to model and pairwise relations between objects.
The term graph here does not refer to data charts such as line graphs, bar charts, histograms and the rest rather, graphs in the context described above are made up of vertices that are connected by edges (which are also called links or lines.) Graphs can also be defined as a mathematical representation of a network and also a description of the relationship that exists between lines and points.
The origin of the graph theory can be traced back to the 18th century when the Swiss mathematician Leonhard Euler solved the mystery of the Konigsberg bridge. The problem of the Konigsberg bridge was based on the possibility of finding a path over every one of the seven parts of the bridge that span a forked river flowing past an island. This puzzle was solved by Euler who argued that such a path did not exist. Although his proof was based only on the physical arrangement of the bridge, it proved to be the first and earliest theory in graph theory.
The meaning of some very important terminologies in the graph is given below. The meanings of these terms will greatly help in the understanding of graph theory.
1. Vertex: A vertex or node in graph theory is a point of circle from which graphs are made.
2. Edge: is a line joining a pair of nodes
3. An Arc: Is a directed line (a pair of ordered vertices)
4. A loop: is an edge or arc that is joining a vertex to itself
5. A walk: is a series of vertices and edges
6. An edge contraction: this can be defined as the process of removing an edge from a graph by merging the two vertices it used to join
7. A graph traversal: is an exploration of a graph in which the vertices are updated one by one
8. A Hamiltonian cycle: this is a closed-loop where every node is visited exactly once.
To understand the concept of graph theory properly, we must get familiar with the various types of graphs that we have. Although there are numerous types of graphs, the common ones are explained below.
• Null graph: Also called the empty graph, the null graph is a type of graph that does not have edges between any of its vertices
• Connected graph: This is a graph in which there is a part of edges between every pair of vertices in the graph.
• Disconnected graph: a graph in which the path of edges does not always connect at every vertex.
• Directed graph: In this type of graph, the edges are always directed by arrows which indicate that the relationship represented by the edges only applies from one point to another and not the other way round
• Undirected graph: This type of graph does not contain any type of arrow to indicate the relationship that exists between the edges of the vertices.
• Bipartite graph: this is a graph that can be split into two sets of vertices such that the edges only goes between sets, not within them.
• Weighted graph: A graph in which weights of numerical values are assigned to each of the edges of the graph.
• Simple graph: This is an undirected graph in which there is at most one edge between each pair of vertices, and there are no loops.
• Multi-graph: a graph in which there are multiple edges to any pair of vertices, or there are edges from one vertex to itself.
• Planar graph: A graph that can be drawn so that all the edges of the graph does not touch each other.
• Non planer graph: This describes a graph that cannot be drawn without at least one pair of its edges crossing each other.
This various types of graphs are often used for different types of situations. They carry out different functions in the graph theory.
The graph theory which is a major aspect of mathematics is used to model many types of relationships and processes that occur in various aspect of the society. Aspects like biology, computer studies, linguistics, social science, and physics make use of the graph theory to solve problems relating to this field. The applications of the graph theory in some of these aspects of human life are explained below.
In biology, the graph theory has helped greatly in the efforts to conserve biological life. A vertex can now be used to represent regions where a certain species exist and the edges used to represent the migration parts or movement of these species. This information has helped biologists to analyze the breeding patterns of various animals or in the tracking of the spread of diseases, parasites and how the changes of the climate can affect the organic life.
Graph theory has proven to be very useful in linguistics since natural language often lends itself well to discreet structure.
in computer science, graphs are used to represent and resent data related to network communication, data organization, computational devices, the flow of computation. The relevance of the graph theory in these aspects of computer studies makes it a very important concept in this field.
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